Author Correction: Iontronic pressure sensor with high sensitivity over ultra-broad linear range enabled by laser-induced gradient micro-pyramids

Ruoxi Yang , Ankan Dutta, Bowen Li, Naveen Tiwari, Wanqing Zhang, Zhenyuan Niu, Yuyan Gao, Daniel Erdely, Xin Xin, Tiejun Li , Huanyu Cheng School of Mechanical Engineering, Hebei University of Technology, Tianjin, 300401, China Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, 16802, USA School of Mechanical Engineering, Hebei University of Science & Technology, Shijiazhuang, 050018, China To whom correspondence should be addressed: E-mail: 001036@hebust.edu.cn and huanyu.cheng@psu.edu


Note 1. Design of micro-pyramids
The pyramid structures with excessive height (e.g., over 1 mm) can be difficult to encapsulate or package. Buckling may also occur for those with a high aspect ratio as discussed in the "Gradient pyramidal microstructures from laser-ablated molds" Section and Supplementary Figure 3b. The pressure-sensing range is directly relevant to the deformation of the microstructure. In general, the microstructures with high aspect ratios (or larger size) are beneficial for the increased sensitivity 1 (or increased sensing range). As reported in the literature, the pressure sensors with a microstructure size smaller than 100 µm often show a sensing range of less than 50 kPa (Supplementary Table 4). In contrast, the sensors with a large microstructure size (e.g., side length bigger than 500 µm) can provide a sensing range of more than 1000 kPa 2, 3 .
As for the pyramidal microstructures, when the L/H ratio (L is the bottom side length and H is the height) is √2, the sensors can exhibit a balanced performance between sensitivity and linearity 4,5,6 .
To imitate pyramid structures and also avoid buckling, we design the gradient structure GPML700 structure with L/H of 1.2 and 2.2, which resulted in linear sensing ranges (for all three ionic liquid concentrations).

Note 2. Theoretical analysis of UPM and GPM at the dielectric/electrode interface to understand the enhanced sensitivity without bending
The normalized cross-section of the contact surface of the pyramid microstructure increases proportionally to the square root of the compression force against the pyramid 7 .
However, the capacitance is directly proportional to the area or square of the crosssection of the contact surface, thus, the capacitance becomes directly proportional to the compressive force.
The linear dependence of capacitance and force thus originates from the non-linear relationship between the cross-section and compressive force given in equation (1). For gradient microstructure, the effective cross-section eff of the contact increases in a cascading order for each new pillar ≤ after exceeding every corresponding force . The corresponding force depends on the gradient of the microstructure, as the gradient increases, more exceeding force is required to start deformation of the th pillar. For example, the corresponding force will be zero for all the pillars for a uniform pillar distribution, whereas, for a gradient pillar distribution, only the exceeding force 1 corresponding to the initially deformed pillar = 1 will be zero, and this force increases with the pillar index < +1 ∀ ≤ . The force deforms each pillar with width as the following relationship The effective cross-section eff is given by Using Cauchy-Schwarz inequality, Thereby, the effective capacitance eff is given by Where ( ) is the number of pillars deformed with < < +1 and average force Therefore, the number of pillars ( ) is also a function of pressure applied ( ).
The slope of effective capacitance eff with pressure or sensitivity is thus given by Where is the proportional constant. As the effective capacitance depends on the square of the effective cross-section, the slope of the capacitance force is upperbounded by the number of pillars N that depends on the pressure .

Note 3. Mathematical model of the electric field distribution
The governing equations for electrostatics in the ionic liquid domain are given as follows: where is the formed free electron surface charge density and is the electric where is the elastic strain energy density that is a function of elastic strain state .
For a compressible neo-Hookean material, the elastic strain energy depends on the elastic volume ratio , Lame parameters , λ, and the first invariant of the elastic right Cauchy-Green deformation tensor 1 . The governing equations for the transport of dilute species in the ionic liquid domain are given as follows: where the concentration c of each species and diffusion constant D contribute to the current . The other contribution comes from the migration of species of charge z and mobility due to the electric potential V. The space charge coupling between electrostatics and the transport of dilute species is given by the following governing equations: index finger, respectively. θ is the rotation of the motor for the robotic finger with θt for the present value. C1setpoint is the desired value and its difference with C1 is noted as error.
Supplementary Fig. 12. The comparison between microstructures obtained from two PMMA templates that were separately created by using the same laser parameters, demonstrating reasonably good consistency in the morphology (e.g., height, outline, and surface topography).